Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Sitae Kim ◽  
Alan B. Palazzolo
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Operating Conditions ◽  
Fluid Film ◽  
Stable Limit ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Response Characteristics ◽  
Computation Efficiency

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

Nonlinear Response Characteristics of a Cracked Rotor in Maneuvering Aircraft

2003 ◽  
Author(s):  
Fu-Sheng Lin ◽  
Guang Meng ◽  
Eric Hahn
Keyword(s):  
Fault Diagnosis ◽  
Constant Velocity ◽  
Nonlinear Response ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Parameter Range ◽  
Cracked Rotor ◽  
Periodic Response ◽  
Response Characteristics ◽  
Cracked Rotor System

This paper investigates numerically the nonlinear response of a simple cracked rotor in moving supports, as may occur in aircraft rotors when the aircraft is maneuvering with constant velocity or acceleration. Of particular interest is the influence of the aircraft climb angle. Results show that the climb angle can markedly affect the parameter range for which the system is stable; and over which there results bifurcation, quasi-periodic response or chaotic response. It is shown that aircraft acceleration can also significantly affect the nonlinear behavior of the cracked rotor system, illustrating the possibility for online rotor crack fault diagnosis.

Nonlinear Dynamic Analysis on Planetary Gears-Rotor System in Geared Turbofan Engines

2019 ◽  
Vol 29 (06) ◽  
pp. 1950076 ◽  
Author(s):  
Lanlan Hou ◽  
Shuqian Cao
Keyword(s):  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Chaotic Motions ◽  
Planetary Gears ◽  
Gear Noise ◽  
Turbofan Engines ◽  
Vibration Responses ◽  
Nonlinear Analytical Model

Rotor fatigue and gear noise triggered by nonlinear vibration are the key concerns in Geared Turbofan (GTF) engine which features a new configuration by introducing planetary gears into low-pressure compressor. A nonlinear analytical model of the GTF planetary gears-rotor system is developed, where the torsional effect of rotor and pivotal parameters from gears are incorporated. The nonlinear behavior of the model can be obtained by focusing on the relative torsional vibration responses between gear and rotor. The torsional nonlinear responses are illustrated with bifurcation diagrams, the largest Lyapunov exponents (LLE), Poincaré maps, phase diagrams and spectrum waterfall. Numerical results reveal that the gears-rotor system exhibits abundant torsional nonlinear behaviors, including multiperiodic, quasi-periodic, and chaotic motions. Furthermore, the roads to chaos via quasi-periodicity, period-doubling scenario, mutation and intermittence are demonstrated. The ring gear stiffness at a low value can propel the system into chaos. The damping may complicate the motion, i.e. the system may enter chaos with increasing damping. These results provide an understanding of undesirable torsional dynamic motion for the GTF engine rotor system and therefore serve as a useful reference for engineers in designing and controlling such system.

Chaos and bifurcation analysis of a flexible rotor supported by short journal bearings with non-linear suspension

2000 ◽  
Vol 214 (7) ◽  
pp. 931-947 ◽  
Author(s):  
H-T Yau ◽  
C-K Chen ◽  
C-L Chen
Keyword(s):  
Chaotic Motion ◽  
Power Spectra ◽  
Rotor System ◽  
Numerical Results ◽  
Operating Conditions ◽  
Flexible Rotor ◽  
Bifurcation And Chaos ◽  
Symmetric Motion ◽  
Bifurcation Phenomenon ◽  
Non Linear

The bifurcation and chaos of the unbalanced response of a bearing-rotor system with non-linear suspension are investigated on the basis of the assumption of an incompressible lubricant together with short bearing approximation. Numerical results show that, owing to the non-linear factors, the trajectory of the journal centre demonstrates steady state symmetric motion even when the trajectory of the bearing centre is in a state of disorder. Poincaré maps, bifurcation diagrams and power spectra are used to analyse the behaviour of the bearing centre in the horizontal and vertical directions under different operating conditions. A unidirectional bifurcation phenomenon is detected in the bearing-rotor system in this study. The fractal dimension concept is used to determine whether the system is in a state of chaotic motion. Numerical results show that the dimension of the bearing centre trajectory is fractal and greater than 2 in some operating conditions. This indicates that the bearing centre is in the state of chaotic motion at these operating conditions.

Cavitation and Air Entrainment Effects on the Response of Squeeze Film Supported Rotors

1990 ◽  
Vol 112 (2) ◽  
pp. 347-353 ◽  
Author(s):  
F. Zeidan ◽  
J. Vance
Keyword(s):  
Nonlinear Response ◽  
Wide Spectrum ◽  
Air Entrainment ◽  
Operating Conditions ◽  
Fluid Film ◽  
Squeeze Film ◽  
Experimental Measurements ◽  
Inertial Effects ◽  
Squeeze Film Dampers ◽  
Rigid Rotors

This paper analyzes the effects of air entrainment and cavitation on the synchronous response of squeeze film supported rigid rotors. The fluid film force coefficients are obtained from experimental measurements corresponding to a wide spectrum of operating conditions. These conditions include regimes in which air entrainment effects are dominant. Other conditions where vapor cavitation and fluid inertial effects are dominant are included for comparison. The effects of air entrainment are shown to produce a nonlinear response representative of a softening spring effect not previously known to exist in squeeze film dampers.

The Motion of Floating Systems: Nonlinear Dynamics in Periodic and Random Waves

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Katrin Ellermann
Keyword(s):  
Dynamical Behavior ◽  
Nonlinear Behavior ◽  
Operating Conditions ◽  
Random Waves ◽  
Multiple Components ◽  
Random Disturbances ◽  
Restoring Forces ◽  
Steady State Responses ◽  
External Forces ◽  
Floating Systems

Floating systems, such as ships, barges, or semisubmersibles, show a dynamical behavior, which is determined by their internal structure and the operating conditions caused by external forces e.g., due to waves and wind. Due to the complexity of the system, which commonly includes coupling of multiple components or nonlinear restoring forces, the response can exhibit inherently nonlinear characteristics. In this paper different models of floating systems are considered. For the idealized case of purely harmonic forcing they all show nonlinear behavior such as subharmonic motion or different steady-state responses at constant operating conditions. The introduction of random disturbances leads to deviations from the idealized case, which may change the overall response significantly. Advantages and limitations of the different mathematical models and the applied numerical techniques are discussed.

A New Method to Find the Forced Response of Nonlinear Systems with Dry Friction

2021 ◽  
Author(s):  
Gregory L. Altamirano ◽  
Meng-Hsuan Tien ◽  
Kiran D'Souza
Keyword(s):  
Dry Friction ◽  
Coulomb Friction ◽  
Nonlinear Response ◽  
Piecewise Linear ◽  
Time Integration ◽  
Nonlinear Behavior ◽  
Forced Response ◽  
New Method ◽  
Analysis Tools ◽  
Compatibility Constraint

Abstract Coulomb friction has an influence on the behavior of numerous mechanical systems. Coulomb friction systems or dry friction systems are nonlinear in nature. This nonlinear behavior requires complex and time demanding analysis tools to capture the dynamics of these systems. Recently, efforts have been made to develop efficient analysis tools able to approximate the forced response of systems with dry friction. The objective of this paper is to introduce a methodology that assists in these efforts. In this method, the piecewise-linear nonlinear response is separated into individual linear responses that are coupled together through compatibility constraint equations. The new method is demonstrated on a number of systems of varying complexity. The results obtained by the new method are validated through the comparison with results obtained by time integration. The computational savings of the new method is also discussed.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

scholarly journals Thin Gas Film Isothermal Condensation in Aerodynamic Bearings

2019 ◽  
Vol 141 (11) ◽  
Author(s):  
Eliott Guenat ◽  
Jürg Schiffmann
Keyword(s):  
High Speed ◽  
Waste Heat ◽  
Load Capacity ◽  
Saturation Pressure ◽  
Operating Conditions ◽  
Fluid Film ◽  
Small Scale ◽  
Vapor Compression ◽  
Density Profiles ◽  
Film Pressure

Abstract High-speed small-scale turbomachinery for waste heat recovery and vapor compression cycles is typically supported on gas-lubricated bearings operating close to the saturation conditions of the lubricant. Under particular conditions, the gas film might locally reach the saturation pressure with potentially hazardous effects on the performance of the gas bearing. The present work introduces a model based on the Reynolds equation and the development of cavitation modeling in liquid-lubricated bearings for condensing gas bearings. The effect of condensation on load capacity and pressure and density profiles is investigated for two one-dimensional bearing geometries (parabolic and Rayleigh step) and varying operating conditions. The results suggest that the load capacity is generally negatively affected if condensation occurs. An experimental setup consisting of a Rayleigh-step gas journal bearing with pressure taps to measure the local fluid film pressure is presented and operated in R245fa in near-saturated conditions. The comparison between the evolution of the fluid film pressure under perfect gas and near saturation conditions clearly suggests the occurrence of condensation in the fluid film. These results are corroborated by the very good agreement with the model prediction.

scholarly journals Dynamic Modelling and Response Characteristics of a Magnetic Bearing Rotor System With Auxiliary Bearings

1995 ◽  
Author(s):  
April M. Free ◽  
George T. Flowers ◽  
Victor S. Trent
Keyword(s):  
Dynamic Modelling ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Rolling Element Bearing ◽  
Test Facility ◽  
Iron Core ◽  
Critical Feature ◽  
Response Characteristics ◽  
Rolling Element ◽  
Auxiliary Bearing

Abstract Auxiliary bearings are a critical feature of any magnetic bearing system. They protect the soft iron core of the magnetic bearing during an overload or failure. An auxiliary bearing typically consists of a rolling element bearing or bushing with a clearance gap between the rotor and the inner race of the support. The dynamics of such systems can be quite complex. It is desired to develop a rotordynamic model which describes the dynamic behavior of a flexible rotor system with magnetic bearings including auxiliary bearings. The model is based upon an experimental test facility. Some simulation studies are presented to illustrate the behavior of the model. In particular, the effects of introducing sideloading from the magnetic bearing when one coil fails is studied. These results are presented and discussed.

Heat Transfer and Thermal Elastic Deformation Analysis on the Piston/Cylinder Interface of Axial Piston Machines

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Matteo Pelosi ◽  
Monika Ivantysynova
Keyword(s):  
Heat Transfer ◽  
Temperature Distribution ◽  
Elastic Deformation ◽  
Hydrodynamic Lubrication ◽  
Operating Conditions ◽  
Fluid Film ◽  
Design Elements ◽  
Piston Cylinder ◽  
Viscous Shear ◽  
Axial Piston

The piston/cylinder interface of swash plate–type axial piston machines represents one of the most critical design elements for this type of pump and motor. Oscillating pressures and inertia forces acting on the piston lead to its micro-motion, which generates an oscillating fluid film with a dynamically changing pressure distribution. Operating under oscillating high load conditions, the fluid film between the piston and cylinder has simultaneously to bear the external load and to seal the high pressure regions of the machine. The fluid film interface physical behavior is characterized by an elasto-hydrodynamic lubrication regime. Additionally, the piston reciprocating motion causes fluid film viscous shear, which contributes to a significant heat generation. Therefore, to fully comprehend the piston/cylinder interface fluid film behavior, the influences of heat transfer to the solid boundaries and the consequent solid boundaries’ thermal elastic deformation cannot be neglected. In fact, the mechanical bodies’ complex temperature distribution represents the boundary for nonisothermal fluid film flow calculations. Furthermore, the solids-induced thermal elastic deformation directly affects the fluid film thickness. To analyze the piston/cylinder interface behavior, considering the fluid-structure interaction and thermal problems, the authors developed a fully coupled simulation model. The algorithm couples different numerical domains and techniques to consider all the described physical phenomena. In this paper, the authors present in detail the computational approach implemented to study the heat transfer and thermal elastic deformation phenomena. Simulation results for the piston/cylinder interface of an existing hydrostatic unit are discussed, considering different operating conditions and focusing on the influence of the thermal aspect. Model validation is provided, comparing fluid film boundary temperature distribution predictions with measurements taken on a special test bench.

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